Let's analyze this.

We are given the function:

$f(x) = \frac{-5x + 3}{5x + 4}$

1. First part: $\lim_{x \to +\infty} f(x)$

As $x \to +\infty$, the highest powers of $x$ dominate both the numerator and the denominator. So, we can simplify:

$f(x) = \frac{-5x + 3}{5x + 4} \approx \frac{-5x}{5x} = -1$

Thus,

$\lim_{x \to +\infty} f(x) = -1$

2. Second part: $\lim_{x \to +\infty} f(f(x))$

We already know $\lim_{x \to +\infty} f(x) = -1$. Now, we calculate $f(-1)$:

$f(-1) = \frac{-5(-1) + 3}{5(-1) + 4} = \frac{5 + 3}{-5 + 4} = \frac{8}{-1} = -8$

So,

$\lim_{x \to +\infty} f(f(x)) = -8$

Do you want further details or have any questions about these steps?

Related questions:

1. How does $x \to -\infty$ affect the limit of $f(x)$?
2. What happens to $f(x)$ when $x = 0$?
3. How do we check for horizontal asymptotes?
4. What is the significance of excluding $\frac{4}{5}$ from the domain?
5. Could we explore the limit as $x \to -\infty$ for $f(f(x))$?

Tip:

When dealing with limits at infinity, simplify the function by focusing on the highest-degree terms.